Elastic deformation of a fluid membrane upon colloid binding

Markus Deserno

doi:10.1103/PhysRevE.69.031903

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Elastic deformation of a fluid membrane upon colloid binding

Markus Deserno^*

Department of Chemistry and Biochemistry, UCLA, 405 Hilgard Ave., Los Angeles, California 90095-1569, USA

^*Present address: Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany.

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Phys. Rev. E 69, 031903 – Published 12 March, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.031903

Abstract

When a colloidal particle adheres to a fluid membrane, it induces elastic deformations in the membrane which oppose its own binding. The structural and energetic aspects of this balance are investigated within the framework of a Helfrich Hamiltonian. Based on the full nonlinear shape equations for the membrane profile, a line of continuous binding transitions and a second line of discontinuous envelopment transitions are found, which meet at an unusual triple point. The regime of low tension is studied analytically using a small gradient expansion, while in the limit of large tension scaling arguments are derived which quantify the asymptotic behavior of phase boundary, degree of wrapping, and energy barrier. The maturation of animal viruses by budding is discussed as a biological example of such colloid-membrane interaction events.

References (63)

H. Lodish, A. Berk, S.L. Zipursky, P. Matsudaira, D. Baltimore, and J. Darnell, Molecular Cell Biology (Freeman&Company, New York, 2000).
H. Garoff, R. Hewson, and D.-J.E. Opstelten, Microbiol. Mol. Biol. Rev. 62, 1171 (1998).
S.B. Sieczkarski and G.R. Whittaker, J. Gen. Virol. 83, 1535 (2002).
H. Garoff and K. Simons, Proc. Natl. Acad. Sci. U.S.A. 71, 3988 (1974).
K. Simons, H. Garoff, and A. Helenius, Sci. Am. 246 (2), 58 (1982).
Y.E. Lu and M. Kielian, J. Virol. 74, 7708 (2000).
O. Boussif, F. Lezoualc’h, M.A. Zanta, M.D. Mergny, D. Scherman, B. Demeneix, and J.-P. Behr, Proc. Natl. Acad. Sci. U.S.A. 92, 7297 (1995).
R.M. Hochmuth, J.-Y. Shao, J. Dai, and M.P. Sheetz, Biophys. J. 70, 358 (1996).
R.M. Henderson and H. Oberleithner, Am. J. Physiol. Renal. Physiol. 278, F689 (2000).
M. Abercrombie, J.E.M. Heaysman, and S.M. Pegrum, Exp. Cell Res. 62, 389 (1970).
A. Caspi, O. Yeger, I. Grosheva, A.D. Bershadsky, and M. Elbaum, Biophys. J. 81, 1990 (2001).
C. Dietrich, M. Angelova, and B. Pouligny, J. Phys. II 7, 1651 (1997).
I. Koltover, J.O. Rädler, and C.R. Safinya, Phys. Rev. Lett. 82, 1991 (1999).
R. Lipowsky and H.-G. Döbereiner, Europhys. Lett. 43, 219 (1998).
A.D. Dinsmore, D.T. Wong, P. Nelson, and A.G. Yodh, Phys. Rev. Lett. 80, 409 (1998).
M. Deserno and W.M. Gelbart, J. Phys. Chem. B 106, 5543 (2002).
L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd ed. (Butterworth-Heinemann, Oxford, 1986), Sec. 12, problem 6.
U. Seifert and R. Lipowsky, Phys. Rev. A 42, 4768 (1990).
R. Lipowsky and U. Seifert, Langmuir 7, 1867 (1991).
U. Seifert, Phys. Rev. Lett. 74, 5060 (1995).
J.O. Rädler, T.J. Feder, H.H. Strey, and E. Sackmann, Phys. Rev. E 51, 4526 (1995).
H. Noguchi and M. Takasu, Biophys. J. 83, 299 (2002).
I. Derényi, F. Jülicher, and J. Prost, Phys. Rev. Lett. 88, 238101 (2002).
A. Boulbitch, Europhys. Lett. 59, 910 (2002).
M. Deserno and T. Bickel, Europhys. Lett. 62, 767 (2003).
C.E. Morris and U. Homann, J. Membr. Biol. 179, 79 (2001).
J.-B. Fournier, A. Ajdari, and L. Peliti, Phys. Rev. Lett. 86, 4970 (2001).
M.A. Peterson, J. Appl. Phys. 57, 1739 (1985).
W. Helfrich, Z. Naturforsch. 28c, 693 (1973).
E. Kreyszig, Differential Geometry (Dover, New York, 1991).
The product of the two principal curvatures is called the “Gaussian curvature.” Its integral over a surface S can be rewritten as a line integral of the geodesic curvature over the boundary $\partial S$ of S (Gauss-Bonnet theorem [30]). It is thus invariant under deformations of S which change neither its boundary nor its topology. Under these conditions the Gaussian part in Eq. (1) only results in an additive constant in the energy which can be ignored in what follows.
F. David and S. Leibler, J. Phys. II 1, 959 (1991).
Note that the stiffness of a cell membrane depends on the length scale one is probing: For deformations above a few hundred nanometers not just the lipid bilayer but also the underlying actin cortex must deform, leading to bending constants of several hundred $k_{B} T$ —see, for instance, R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, and E. Sackmann, Biophys. J. 74, 514 (1998). However, on length scales of only a few tens of nanometers, which do not exceed the mesh size of the cortex, the lipid bilayer can deform independently of the actin network and will be characterized by its own, much softer modulus. Note that, for instance, the spectrin network of human erythrocytes has a mesh size of about 75 nm [see D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)].
In Ref. [14] it is suggested that for partially wrapped spheres the deformation of the membrane along the contact line will contribute an additional bending term to the contact line energy which acts as an energy barrier for the encapsulation process and which should scale $\sim \sqrt{z (2 - z)} .$ However, if the membrane is tensionless, such a term is rigorously absent (due to its catenoid shape). Under nonvanishing tension there is indeed a barrier, but it does not have the proposed form, since the bending energy is generally not localized at the rim.
U. Seifert, K. Berndl, and R. Lipowsky, Phys. Rev. A 44, 1182 (1991).
F. Jülicher and U. Seifert, Phys. Rev. E 49, 4728 (1994).
U. Seifert, Adv. Phys. 46, 13 (1997).
Observe that the stronger requirement $h (s) \to const$ is not used. In fact, for zero tension the solution of the shape equations is the catenoid, which diverges logarithmically in the limit $\vec{s} \infty .$ However, this is a special case, and for any nonzero tension the profile height $h (s)$ remains bounded and converges toward a constant.
R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.
In several cases of viral budding the nucleoprotein capsid is not preformed but aggregates simultaneously with the budding process [2]. It then imposes a similar boundary condition for the adhering membrane as a complete capsid, except that the membrane is not prevented from having a contact curvature larger than $1 / a$ (there is no capsid yet to exclude this region of space from being occupied). In this case the states with ${\dot{ψ}}_{0} > 1 / a$ are no longer inaccessible and may become physically relevant.
In this paper “low tension” refers to the regime $\tilde{σ} ≲ 1.$ But just because tension is subdominant to curvature, this by no means implies that it is also small compared to the thermal energy $k_{B} T .$ In fact, at the crossover point $\tilde{σ} = 1,$ the tension energy scales like $E_{ten} \sim σ a^{2} = κ,$ and $κ$ is usually large compared to $k_{B} T .$
Recall that the adhesion energy contributes a linear term $- \tilde{w} z$ to the total energy of the complex, see Eq. (4). Increasing $\tilde{w}$ hence implies “pulling the energy curve $\tilde{E} (z)$ down at the right-hand side,” thus the enveloped state becomes increasingly favorable.
B. Fourcade, L. Miao, M. Rao, M. Wortis, and R.K.P. Zia, Phys. Rev. E 49, 5276 (1994).
As the neck contracts, the membrane deformation will ultimately occur on length scales small compared to the typical size $λ$ introduced in Eq. (2), hence tension ceases to be important. But if only bending counts, the optimal solution is the catenoid, which has zero bending energy. Indeed, in the limit $\vec{z} 2$ the meridinal contact curvature ${\dot{ψ}}_{0}$ converges toward $- 1 / a,$ i.e., the negative of the azimuthal contact curvature. Of course, once the deformation of the membrane occurs on length scales comparable to its thickness (typically a few nanometers), neither linear elasticity theory underlying Eq. (1) nor the description of the membrane as an idealized surface are appropriate, and the bilayer structure needs to be modeled in more detail.
The negative slope of $\tilde{E} (z)$ at $z = 2$ [see inset of Fig. 33] implies that the regions of the bilayer at the upper neck are pushed against each other. This may potentially facilitate a subsequent fusion event.
J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Dover, New York, 2002).
U. Seifert and R. Lipowsky, in Structure and Dynamics of Membranes, edited by R. Lipowsky and E. Sackmann, Handbook of Biological Physics, Vol. 1A (Elsevier, New York/North-Holland, Amsterdam, 1995).
Equation (17) can also be derived by a direct small gradient approximation of the shape equations (9). This approach is less elegant, but it illustrates quite vividly how many “unpleasant” terms drop out in lowest order—and only in lowest order. Hence, extending the small gradient expansion to the next order appears about as hard as treating the full nonlinear problem.
Handbook of Mathematical Functions, 9th ed., edited by M. Abramowitz and I.A. Stegun (Dover, New York, 1970).
R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, Adv. Comput. Math. 5, 329 (1996).
The (essentially) logarithmic divergence is obvious by looking at the inverse of $W_{- 1} (x),$ namely, the function ${xe}^{x},$ which approaches 0 (essentially) exponentially from the negative side as x approaches -∞. A more precise statement can be obtained by making use of the functional relation $W_{- 1} (x) = \ln | x | - \ln | W_{- 1} (x) |,$ which can be iterated to get increasingly accurate expressions for $W_{- 1} (x) .$ Since ${limop}_{\vec{x} 0^{-}} W_{- 1} (x) / \ln | x | = 1,$ the lowest order reads $W_{- 1} (x) \approx \ln | x |,$ and the next correction is then given by $W_{- 1} (x) \approx \ln | x | - \ln | \ln | x | | .$
C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (Springer, New York, 1999).
One more bit of support stems from Eq. (13), which suggests that $a {\dot{ψ}}_{0}$ and $\sqrt{1 + 2 \tilde{σ} z}$ are of the same order. In fact, for $z = 1$ one rigorously has $| a {\dot{ψ}}_{0} | = \sqrt{1 + 2 \tilde{σ}},$ for otherwise $p_{r} (0)$ would be infinite.
In the absence of a theoretical description of the barrier height in the low tension regime (a small gradient expansion is not enough), the exponent 0.86 is of course nothing but a purely empirical description of the numerical results, which are equally compatible with the form $a_{1} \tilde{σ} \times \log {(a}_{2} \tilde{σ}) .$
S. Leibler, J. Phys. (Paris) 47, 507 (1986).
T. Kawakatsu, D. Andelman, K. Kawasaki, and T. Taniguchi, J. Phys. II 3, 971 (1993); ibid.T. Taniguchi, K. Kawasaki, D. Andelman, and T. Kawakatsu, 4, 1333 (1994).
U. Seifert, Phys. Rev. Lett. 70, 1335 (1993).
F. Jülicher and R. Lipowsky, Phys. Rev. E 53, 2670 (1996).
N. Chazal and D. Gerlier, Microbiol. Mol. Biol. Rev. 67, 226 (2003).
S. Tzlil, M. Deserno, W.M. Gelbart, and A. Ben-Shaul, Biophys. J. (to be published).
The only subtle point is that the arclength s is varied as well, which, for instance, implies that the Hamiltonian H enters into Eq. (A2), and not the Lagrangian L. See, for instance, the discussion in H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Chap. 8–6.
R. Rosso and E.G. Virga, Proc. R. Soc. London, Ser. A 455, 4145 (1999).
R. Capovilla and J. Guven, Phys. Rev. E 66, 041604 (2002).

[c1] H. Lodish, A. Berk, S.L. Zipursky, P. Matsudaira, D. Baltimore, and J. Darnell, Molecular Cell Biology (Freeman&Company, New York, 2000).

[c2] H. Garoff, R. Hewson, and D.-J.E. Opstelten, Microbiol. Mol. Biol. Rev. 62, 1171 (1998).

[c3] S.B. Sieczkarski and G.R. Whittaker, J. Gen. Virol. 83, 1535 (2002).

[c4] H. Garoff and K. Simons, Proc. Natl. Acad. Sci. U.S.A. 71, 3988 (1974).

[c5] K. Simons, H. Garoff, and A. Helenius, Sci. Am. 246 (2), 58 (1982).

[c6] Y.E. Lu and M. Kielian, J. Virol. 74, 7708 (2000).

[c7] O. Boussif, F. Lezoualc’h, M.A. Zanta, M.D. Mergny, D. Scherman, B. Demeneix, and J.-P. Behr, Proc. Natl. Acad. Sci. U.S.A. 92, 7297 (1995).

[c8] R.M. Hochmuth, J.-Y. Shao, J. Dai, and M.P. Sheetz, Biophys. J. 70, 358 (1996).

[c9] R.M. Henderson and H. Oberleithner, Am. J. Physiol. Renal. Physiol. 278, F689 (2000).

[c10] M. Abercrombie, J.E.M. Heaysman, and S.M. Pegrum, Exp. Cell Res. 62, 389 (1970).

[c11] A. Caspi, O. Yeger, I. Grosheva, A.D. Bershadsky, and M. Elbaum, Biophys. J. 81, 1990 (2001).

[c12] C. Dietrich, M. Angelova, and B. Pouligny, J. Phys. II 7, 1651 (1997).

[c13] I. Koltover, J.O. Rädler, and C.R. Safinya, Phys. Rev. Lett. 82, 1991 (1999).

[c14] R. Lipowsky and H.-G. Döbereiner, Europhys. Lett. 43, 219 (1998).

[c15] A.D. Dinsmore, D.T. Wong, P. Nelson, and A.G. Yodh, Phys. Rev. Lett. 80, 409 (1998).

[c16] M. Deserno and W.M. Gelbart, J. Phys. Chem. B 106, 5543 (2002).

[c17] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd ed. (Butterworth-Heinemann, Oxford, 1986), Sec. 12, problem 6.

[c18] U. Seifert and R. Lipowsky, Phys. Rev. A 42, 4768 (1990).

[c19] R. Lipowsky and U. Seifert, Langmuir 7, 1867 (1991).

[c20] U. Seifert, Phys. Rev. Lett. 74, 5060 (1995).

[c21] J.O. Rädler, T.J. Feder, H.H. Strey, and E. Sackmann, Phys. Rev. E 51, 4526 (1995).

[c22] H. Noguchi and M. Takasu, Biophys. J. 83, 299 (2002).

[c23] I. Derényi, F. Jülicher, and J. Prost, Phys. Rev. Lett. 88, 238101 (2002).

[c24] A. Boulbitch, Europhys. Lett. 59, 910 (2002).

[c25] M. Deserno and T. Bickel, Europhys. Lett. 62, 767 (2003).

[c26] C.E. Morris and U. Homann, J. Membr. Biol. 179, 79 (2001).

[c27] J.-B. Fournier, A. Ajdari, and L. Peliti, Phys. Rev. Lett. 86, 4970 (2001).

[c28] M.A. Peterson, J. Appl. Phys. 57, 1739 (1985).

[c29] W. Helfrich, Z. Naturforsch. 28c, 693 (1973).

[c30] E. Kreyszig, Differential Geometry (Dover, New York, 1991).

[c31] The product of the two principal curvatures is called the “Gaussian curvature.” Its integral over a surface S can be rewritten as a line integral of the geodesic curvature over the boundary $\partial S$ of S (Gauss-Bonnet theorem [30]). It is thus invariant under deformations of S which change neither its boundary nor its topology. Under these conditions the Gaussian part in Eq. (1) only results in an additive constant in the energy which can be ignored in what follows.

[c32] F. David and S. Leibler, J. Phys. II 1, 959 (1991).

[c33] Note that the stiffness of a cell membrane depends on the length scale one is probing: For deformations above a few hundred nanometers not just the lipid bilayer but also the underlying actin cortex must deform, leading to bending constants of several hundred $k_{B} T$ —see, for instance, R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, and E. Sackmann, Biophys. J. 74, 514 (1998). However, on length scales of only a few tens of nanometers, which do not exceed the mesh size of the cortex, the lipid bilayer can deform independently of the actin network and will be characterized by its own, much softer modulus. Note that, for instance, the spectrin network of human erythrocytes has a mesh size of about 75 nm [see D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)].

[c34] In Ref. [14] it is suggested that for partially wrapped spheres the deformation of the membrane along the contact line will contribute an additional bending term to the contact line energy which acts as an energy barrier for the encapsulation process and which should scale $\sim \sqrt{z (2 - z)} .$ However, if the membrane is tensionless, such a term is rigorously absent (due to its catenoid shape). Under nonvanishing tension there is indeed a barrier, but it does not have the proposed form, since the bending energy is generally not localized at the rim.

[c35] U. Seifert, K. Berndl, and R. Lipowsky, Phys. Rev. A 44, 1182 (1991).

[c36] F. Jülicher and U. Seifert, Phys. Rev. E 49, 4728 (1994).

[c37] U. Seifert, Adv. Phys. 46, 13 (1997).

[c38] Observe that the stronger requirement $h (s) \to const$ is not used. In fact, for zero tension the solution of the shape equations is the catenoid, which diverges logarithmically in the limit $\vec{s} \infty .$ However, this is a special case, and for any nonzero tension the profile height $h (s)$ remains bounded and converges toward a constant.

[c39] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

[c40] In several cases of viral budding the nucleoprotein capsid is not preformed but aggregates simultaneously with the budding process [2]. It then imposes a similar boundary condition for the adhering membrane as a complete capsid, except that the membrane is not prevented from having a contact curvature larger than $1 / a$ (there is no capsid yet to exclude this region of space from being occupied). In this case the states with ${\dot{ψ}}_{0} > 1 / a$ are no longer inaccessible and may become physically relevant.

[c41] In this paper “low tension” refers to the regime $\tilde{σ} ≲ 1.$ But just because tension is subdominant to curvature, this by no means implies that it is also small compared to the thermal energy $k_{B} T .$ In fact, at the crossover point $\tilde{σ} = 1,$ the tension energy scales like $E_{ten} \sim σ a^{2} = κ,$ and $κ$ is usually large compared to $k_{B} T .$

[c42] Recall that the adhesion energy contributes a linear term $- \tilde{w} z$ to the total energy of the complex, see Eq. (4). Increasing $\tilde{w}$ hence implies “pulling the energy curve $\tilde{E} (z)$ down at the right-hand side,” thus the enveloped state becomes increasingly favorable.

[c43] B. Fourcade, L. Miao, M. Rao, M. Wortis, and R.K.P. Zia, Phys. Rev. E 49, 5276 (1994).

[c44] As the neck contracts, the membrane deformation will ultimately occur on length scales small compared to the typical size $λ$ introduced in Eq. (2), hence tension ceases to be important. But if only bending counts, the optimal solution is the catenoid, which has zero bending energy. Indeed, in the limit $\vec{z} 2$ the meridinal contact curvature ${\dot{ψ}}_{0}$ converges toward $- 1 / a,$ i.e., the negative of the azimuthal contact curvature. Of course, once the deformation of the membrane occurs on length scales comparable to its thickness (typically a few nanometers), neither linear elasticity theory underlying Eq. (1) nor the description of the membrane as an idealized surface are appropriate, and the bilayer structure needs to be modeled in more detail.

[c45] The negative slope of $\tilde{E} (z)$ at $z = 2$ [see inset of Fig. 33] implies that the regions of the bilayer at the upper neck are pushed against each other. This may potentially facilitate a subsequent fusion event.

[c46] J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Dover, New York, 2002).

[c47] U. Seifert and R. Lipowsky, in Structure and Dynamics of Membranes, edited by R. Lipowsky and E. Sackmann, Handbook of Biological Physics, Vol. 1A (Elsevier, New York/North-Holland, Amsterdam, 1995).

[c48] Equation (17) can also be derived by a direct small gradient approximation of the shape equations (9). This approach is less elegant, but it illustrates quite vividly how many “unpleasant” terms drop out in lowest order—and only in lowest order. Hence, extending the small gradient expansion to the next order appears about as hard as treating the full nonlinear problem.

[c49] Handbook of Mathematical Functions, 9th ed., edited by M. Abramowitz and I.A. Stegun (Dover, New York, 1970).

[c50] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, Adv. Comput. Math. 5, 329 (1996).

[c51] The (essentially) logarithmic divergence is obvious by looking at the inverse of $W_{- 1} (x),$ namely, the function ${xe}^{x},$ which approaches 0 (essentially) exponentially from the negative side as x approaches -∞. A more precise statement can be obtained by making use of the functional relation $W_{- 1} (x) = \ln | x | - \ln | W_{- 1} (x) |,$ which can be iterated to get increasingly accurate expressions for $W_{- 1} (x) .$ Since ${limop}_{\vec{x} 0^{-}} W_{- 1} (x) / \ln | x | = 1,$ the lowest order reads $W_{- 1} (x) \approx \ln | x |,$ and the next correction is then given by $W_{- 1} (x) \approx \ln | x | - \ln | \ln | x | | .$

[c52] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (Springer, New York, 1999).

[c53] One more bit of support stems from Eq. (13), which suggests that $a {\dot{ψ}}_{0}$ and $\sqrt{1 + 2 \tilde{σ} z}$ are of the same order. In fact, for $z = 1$ one rigorously has $| a {\dot{ψ}}_{0} | = \sqrt{1 + 2 \tilde{σ}},$ for otherwise $p_{r} (0)$ would be infinite.

[c54] In the absence of a theoretical description of the barrier height in the low tension regime (a small gradient expansion is not enough), the exponent 0.86 is of course nothing but a purely empirical description of the numerical results, which are equally compatible with the form $a_{1} \tilde{σ} \times \log {(a}_{2} \tilde{σ}) .$

[c55] S. Leibler, J. Phys. (Paris) 47, 507 (1986).

[c56] T. Kawakatsu, D. Andelman, K. Kawasaki, and T. Taniguchi, J. Phys. II 3, 971 (1993); ibid.T. Taniguchi, K. Kawasaki, D. Andelman, and T. Kawakatsu, 4, 1333 (1994).

[c57] U. Seifert, Phys. Rev. Lett. 70, 1335 (1993).

[c58] F. Jülicher and R. Lipowsky, Phys. Rev. E 53, 2670 (1996).

[c59] N. Chazal and D. Gerlier, Microbiol. Mol. Biol. Rev. 67, 226 (2003).

[c60] S. Tzlil, M. Deserno, W.M. Gelbart, and A. Ben-Shaul, Biophys. J. (to be published).

[c61] The only subtle point is that the arclength s is varied as well, which, for instance, implies that the Hamiltonian H enters into Eq. (A2), and not the Lagrangian L. See, for instance, the discussion in H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Chap. 8–6.

[c62] R. Rosso and E.G. Virga, Proc. R. Soc. London, Ser. A 455, 4145 (1999).

[c63] R. Capovilla and J. Guven, Phys. Rev. E 66, 041604 (2002).

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Elastic deformation of a fluid membrane upon colloid binding

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